Estimating the effects of lockdown timing on COVID-19 cases and deaths in England: A counterfactual modelling study

Background During the first wave of the COVID-19 pandemic, the United Kingdom experienced one of the highest per-capita death tolls worldwide. It is debated whether this may partly be explained by the relatively late initiation of voluntary social distancing and mandatory lockdown measures. In this study, we used simulations to estimate the number of cases and deaths that would have occurred in England by 1 June 2020 if these interventions had been implemented one or two weeks earlier, and the impact on the required duration of lockdown. Methods Using official reported data on the number of Pillar 1 lab-confirmed cases of COVID-19 and associated deaths occurring in England from 3 March to 1 June, we modelled: the natural (i.e. observed) growth of cases, and the counterfactual (i.e. hypothetical) growth of cases that would have occurred had measures been implemented one or two weeks earlier. Under each counterfactual condition, we estimated the expected number of deaths and the time required to reach the incidence observed under natural growth on 1 June. Results Introducing measures one week earlier would have reduced by 74% the number of confirmed COVID-19 cases in England by 1 June, resulting in approximately 21,000 fewer hospital deaths and 34,000 fewer total deaths; the required time spent in full lockdown could also have been halved, from 69 to 35 days. Acting two weeks earlier would have reduced cases by 93%, resulting in between 26,000 and 43,000 fewer deaths. Conclusions Our modelling supports the claim that the relatively late introduction of social distancing and lockdown measures likely increased the scale, severity, and duration of the first wave of COVID-19 in England. Our results highlight the importance of acting swiftly to minimise the spread of an infectious disease when case numbers are increasing exponentially.

1. NHS England, which reports figures on deaths occurring in hospitals for which the patient either tested positive for COVID-19 or where COVID-19 was mentioned on the death certificate. 4 These data are available as an Excel file under the heading 'Data' and subheading 'COVID-19 daily announced deaths' of the referenced webpage. 4 Updated datasets are published every day, meaning that substantial changes are likely to occur for counts reported during the previous 5 days and more minor changes may occur further in the past.

Office for National Statistics (ONS)
, which reports figures on all registered deaths for which COVID-19 was mentioned on the death certificate. 5 These data are available as an Excel file on the referenced webpage. 5 Updated datasets are published weekly.
For both sources, deaths are attributed to the date of death, rather than the date of reporting/announcement.
NHS England data were downloaded on 22 July 2020, which contained data up to 20 July 2020 [source filename: COVID-19-total-announced-deaths-21-July-2020.xlsx]. Daily death figures were taken from the tab entitled 'Tab1 Deaths By Region', which contains incident deaths for the whole of England and disaggregated by region. Cumulative death figures were calculated by the authors while preparing the data for subsequent analysis.
ONS data were downloaded on 22 July 2020, which contained data up to and including the 28 th week of 2020, i.e. up to 10 July 2020 [source filename: publishedweek282020.xlsx]. Cumulative death figures were taken from the tab entitled 'Covid-19 -England comparisons', column 'ONS deaths by actual data of death -registered by 18 July'. Daily death figures were calculated by the authors while preparing the data for subsequent analysis.

Identification of growth periods and parameters
We considered the period between 3 March (i.e. = 1) and 1 June (i.e. = ).
We allowed for a lag of up to 21 days from when social distancing measures began (i.e. 17 March) and when their effects on the growth rate became visible (i.e. knot date ). Similarly, we allowed for a lag of up to 21 days from when lockdown measures began (i.e. 23 March) and when their effects on the growth rate became visible (i.e. knot date ). We considered all possible pairs of knot dates ( , ) for which < .
For each pair of candidate knot dates ( , ), we fit the following spline model: We accounted for dependencies between observations and accommodated apparent 'weekend effects' in data collection (i.e. where fewer COVID-19 tests are administered on the weekends) by fitting an Arima model with one autoregressive term (i.e. p = 1) and seven-day seasonal adjustment. 6 For each pair of candidate knot dates, we recorded the three growth factor corresponding to each of the three spline segments as the slope of the segment (i.e. 1 , 1 , or 1 ) plus one, since We also recorded the standard deviation (SD) of each growth factor as the estimated standard error for each of 1 , 1 , or 1 , respectively. Note that we did not evaluate the spline model directly (e.g. via AIC/BIC), since these measures were determined to be poor calibration statistics. Because of the way that cases increase multiplicatively (i.e. by a factor of ), small overestimations in the growth rate in the first period correspond to large errors in model predictions (a manifestation of the 'butterfly effect'), which AIC/BIC cannot account for.
Instead, we estimated how well the given knot dates and associated growth factors predicted the observed growth of cases between 3 March and 1 June. For each day , 1 ≤ ≤ , the estimated growth factor was applied the number of incident cases on the previous day according to the period of growth in which it fell, in order to calculate the number of incident cases on the current day. The Poisson deviance between the observed (7-day moving average) and predicted incident and cumulative cases over the entire period was calculated; 7 this criteria was selected due to it being a likelihood-based measure that can account for the fact that the model error likely scales with the number of cases. 7 The 10 'best' pairs of knot points according to each of the following two criteria were calculated: (1) those which produced the lowest Poisson deviance with respect to incident cases, and (2) those which produced the lowest Poisson deviance with respect to cumulative cases. Pairs of knot points which satisfied both criteria were retained. For each of these pairs, we constructed a likelihood-based probability of that pair based on its Poisson deviance with respect to cumulative cases, since all knot point pairs do not fit the data equally well. Because lower deviance values reflect higher likelihood, we took the inverse for each knot point pair and rescaled them so that all values summed to 1.

Calculation of case fatality ratios (CFRs)
The case fatality ratio (CFR) for COVID-19 is defined as the proportion of deaths attributable to COVID-19 among those diagnosed with COVID-19 over a given time period. For example, the CFR on day can be expressed as ℎ ⁄ .
We calculated two separate CFRs on June 1 (i.e. = ) according to two different death counts: . This CFR considers COVID-19-related deaths occurring in hospitals (data from NHS England; 4 see pages 1-2).
We have not adjusted our CFR estimates since they are unlikely to deviate a substantial amount from the true estimates. Figure 1 displays each of 1 and 2 over time, in which it is apparent that the estimates have either begun to or have already levelled off by 1 June. Our estimates are therefore conservative but unlikely to be substantially biased.
We did not take into account variation/uncertainty in the CFRs.

Figure 1
Case fatality ratios (CFRs) in England over time, as calculated using deaths across all settings and hospital deaths only.

Sensitivity analyses
This section describes two sensitivity analyses which were performed.

Inclusion of cases identified by both Pillar 1 and Pillar 2
We explored the sensitivity of our results to the exclusion COVID-19 cases identified by Pillar 2 testing by re-running all analyses using cases identified by both Pillar 1 and Pillar 2 of the British government's testing programme. Table 1 summarises the knot date pairs deemed most likely by this analysis, and Figure 2 displays the observed relationship between cumulative and daily number of new lab-confirmed cases (Pillar 1 and Pillar 2) in England from 30 January to 1 June. The spline models corresponding to the knot date pairs in Table 1 Figure 4 show the incident and cumulative lab-confirmed cases of COVID-19 under each stochastically-simulated growth scenario, overlaying the observed data. The results of these simulations are given in Table 2 and Table 3.
These simulations suggest that implementing social distancing and lockdown one or two weeks earlier would have resulted in a 72% or 92% reduction in the total number of cases, respectively, by 1 June. Under the natural history, our model estimated 1897 incident cases on 1 June; this threshold was not exceeded in either counterfactual history. The Poisson deviance of the natural growth model with respect to incident cases is 5,143, and 5,915 with respect to cumulative cases.
As is evident from these results, the inclusion of Pillar 2 data produces slightly more conservative estimates of percentage reductions in COVID-19 cases numbers, but these remain broadly in line with results from our primary analysis. Additionally, the inclusion of Pillar 2 data creates less well-defined periods of growth, leading to a poorer fitting model overall.   1 , which utilises data from NHS England on deaths occurring in hospitals; 4 and 2 , which utilises data from ONS on all deaths. 5 The mean number of deaths from 100,000 simulation runs are given for the three modelled scenarios, with 95% simulation intervals (i.e.

Equal probabilities of best knot date pairs
We explored the sensitivity of our results to the construction of our likelihood-based probability of each knot point pair, by conducting the stochastic simulation in which the probability of each knot point pair was equal (i.e. = 0.2).
The results of this simulation are given in Table 4 and Table 5. As is evident, the simulation intervals (SIs) for both cases and deaths on 1 June are slightly wider than in the primary analysis, but bottom-line inferences remain unchanged. The Poisson deviance of the natural growth model with respect to incident cases is 2,408, and 2,360 with respect to cumulative cases.  Cumulative number of deaths resulting from COVID-19 in England on 1 June for each scenario modelled in the sensitivity analysis. Estimates are given according to two separate case fatality ratios: 1 , which utilises data from NHS England on deaths occurring in hospitals; 4 and 2 , which utilises data from ONS on all deaths. 5 The mean number of deaths from 100,000 simulation runs are given for the three modelled scenarios, with 95% simulation intervals (i.e.

Code
All simulation and analytical code can be accessed at https://github.com/KFArnold/covidcounterfactual.
A README.md file is available in the repository, which gives a broad overview of the repository's structure and usage. We also provide detailed notes here regarding how the code functions.
The repository has four main folders: 1. Data: This folder contains all cases and deaths data that are used for the analysis. The Code folder contains four scripts, the main functions of each of which are described below: